Tectonophysics 458 (2008) 96–104

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Tectonophysics

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Strain localisation and weakening of the lithosphere during extension

Klaus Regenauer-Lieb a,b,1, Gideon Rosenbaum c,⁎, Roberto F. Weinberg d,2

a School of Earth and Geographical Sciences, The University of Western Australia, Perth, Western Australia 6009, Australiab CSIRO Exploration and Mining, Kensington, Western Australia 6151, Australiac School of Physical Sciences, University of Queensland, Brisbane, Queensland 4072, Australiad School of Geosciences, Monash University, Clayton, Victoria 3800, Australia

⁎ Corresponding author. Fax: +61 7 3365 1277.E-mail addresses: klaus@cyllene.uwa.edu.au (K. Rege

g.rosenbaum@uq.edu.au (G. Rosenbaum), Roberto.Wein(R.F. Weinberg).

1 Fax: +61 8 6488 1050.2 Fax: +61 3 9905 4903.

0040-1951/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.tecto.2008.02.014

A B S T R A C T

A R T I C L E I N F O

Article history:

We explore the sensitivity o Received 1 September 2007Received in revised form 25 February 2008Accepted 27 February 2008Available online 18 March 2008

Keywords:ExtensionNumerical modellingDetachment faultingRheologyLithospheric strength

f extensional systems to the thermal structure of the lithosphere using numericalsimulations that fully couple the energy, momentum and continuity equations. The rheology of thelithosphere is controlled by weakening processes, such as shear heating, that localises strain into shear zonesand faults. Numerical models show that during extension of an initially unpatterned lithosphere, structuresdevelop spontaneously out of basic thermodynamic energy fluxes, and without the imposition of ad hoc ruleson strain localisation. This contrasts with the classical Mohr–Coulomb theory for brittle localisation, whichprescribes the angles of faults by a mathematical rule. Our results show that the mode of extension issensitive to subtle changes in rheology, heat flux and geometry of the system. This sensitivity lies at the coreof the variety and complexity observed in extensional systems. Localisation processes make the lithosphereweaker than previously estimated from the Brace–Goetze quasi-static approach. Consequently, typicalestimates for plate tectonic forces are capable of splitting the lithosphere under extension, even without therole of active magmatism.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

The strength of the lithosphere is one of the most fundamentalparameters for understanding lithospheric extension. Yet our under-standing of lithospheric strength is relatively rudimentary, anddescriptions of the rheology of the lithosphere are a matter of debate.The current modelling paradigm is based on the concept of the Brace–Goetze strength profile (Goetze and Evans, 1979; Brace and Kohlstedt,1980), informally known as the ‘Christmas Tree’. It assumes a com-bination of frictional sliding and creep mechanisms, and is typicallycharacterised by a weak lower crust sandwiched between the strongbrittle–ductile transition area and the strong mantle (dashed curve inFig. 1). The Brace–Goetze profile provides a quasi-static view of litho-spheric strength and does not consider the role of weakeningprocesses that localise strain. Consequently, Brace–Goetze profiles orother simplifications usually yield over-estimates of the strength of thelithosphere (Kusznir and Park, 1984; Ranalli and Murphy, 1987; Buck,2006).

In this paper we describe an alternative approach for estimatinglithospheric strength by considering energy fluxes in a system that

nauer-Lieb),berg@sci.monash.edu.au

l rights reserved.

fully couples the energy,momentumand continuityequations. Such anapproach takes into account brittle andductile energy feedback effects,which arise from temperature and pressure perturbations, and canlead to runaway processes of strain localisation (Regenauer-Lieb andYuen, 2003; Regenauer-Lieb et al., 2006). We show that the strength ofthe lithosphere during extension is subjected to temporal and spatialchanges. This allows the development of elastic zones bounded byweak zones, simulating typical core complex and detachmentstructures recognised in extensional environments (Rosenbaumet al., 2005; Regenauer-Lieb et al., 2006; Weinberg et al., 2007).

2. The quasi-static Brace–Goetze strength profile of the lithosphere

The classic approach to estimating the strength of the lithosphere,known as the Brace–Goetze profile, relies on a combination ofpressure-dependent brittle layer overlying a temperature- and strainrate-dependent viscous layer (Goetze and Evans,1979; Kohlstedt et al.,1995). The strength of the brittle layer is defined by Byerlee's (1978)law,which assumes a linear relationship between the shear strength ofrocks and the pressure, with a constant friction coefficient. The latter iscommonly assumed to be independent of temperature and material.The strength of the ductile layer is derived from steady-state creepexperiments (Ranalli, 1995), and is most commonly described by apower law relationship between the stress and the strain rate.Commonly, a fixed strain rate is assumed (typically 10−15–10−14 s−1),allowing the definition of the strength of the whole lithosphere as anintegral quantity.

Fig. 1. A classical Brace–Goetze (“Christmas Tree”) yield strength envelope (dashed line)shownagainst a dynamic elasto-visco-plastic strengthprofile (black line) (afterRegenauer-Lieb et al., 2006). The dynamic profile evolves through time and is calculated here shortlyafter the onset of extension (130 kyr). The lithosphere is made up of a 42 km crust (quartzrheology) overlying an olivinemantle. The dynamic strength profile is not an average valuefor each depth, but is shown for a single 1D vertical line that intersects with twodetachment faults in the 10–20 km depth range. Note that the energy feedbacks of the toppurely brittle layer (down to ~2 kmdepths) are not considered, because at shallow depths,the surface energy of cracks, which is not incorporated in the current numerical model,becomes increasingly important. The shape of this profile changes from place to place andin time. Our particular choice of 1D profile represents one of the weakest profiles in thecross section and showsminimum strength values for the top brittle crust and for the areabelow 52 km, due to unloading of stresses by nearby shear zones. The maximum strengthanywhere in the model is limited by the Brace–Goetze profile. Therefore, other dynamicstrengths profiles in 2D and 3D will be restricted to the grey area.

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In the context of the Brace–Goetze strength profile, the brittle–ductile transition is defined by the intersection of the brittle and theductile strength curves. The material above the intersection isassumed to behave in a brittle manner, whereas the material belowis assumed to flow viscously. However, this sharp brittle–ductileboundary overlooks any semi-brittle/semi-ductile behaviour such asobserved in laboratory experiments and nature (Kohlstedt et al., 1995).This shortcoming remains unresolved even in more sophisticatednumerical approaches that derive local strain rate from a momentumbalance rather than assuming a constant strain rate everywhere (e.g.Albert et al., 2000).

The Brace–Goetze strength profile is an important concept thatprovides a first-order explanation for the geodynamics of plates.However, it does not adequately account for the dynamic evolution ofdeformation. In particular, it does not consider the complex commu-nication of stresses and temperature (energy) between the plasticallyand viscously deforming parts of the system, or the feedback effectsthat lead to strain localisation and weakening of the system that isregulated by energy flux. The result of these simplifications is that itgenerates a region of maximum strength at the brittle–ductiletransition in the crust, and also in the mantle, depending on thetemperature profile (e.g. Ranalli and Murphy, 1987). However,geological evidence from extensional environments shows that strainis localised roughly at the crustal brittle–ductile transition (Davis andConey, 1979; Miller et al., 1983; Wernicke, 1992; Jolivet et al., 1998;

Chéry, 2001), an observation that is not reconciled with its supposedmaximum crustal strength (Gueydan et al., 2004). In addition, basedon seismological studies, there is no convincing evidence for theoccurrence of seismic activity in the continental mantle, as would beexpected from the Brace–Goetze strength profile (Maggi et al., 2000).

The concept of time dependent strength evolution of the litho-sphere was already introduced in the 1980s (Kusznir, 1982; Kusznirand Park, 1982, 1984), but explicit dynamic coupling was notconsidered. These papers demonstrated that, by considering elasticity,the solution to lithospheric extension changes dramatically. In thesimple case of constant applied load, creep in the lower crust amplifiesthe stresses in a strong, essentially elastic layer at the brittle–ductiletransition to the extent that a low applied constant force can leadover time towhole scale lithosphere failure. Kusznir (1982) coined theterm ‘visco-elastic stress amplification’ to describe this importantphenomenon.

In numerical models that do not consider the energy flux throughthe system, strain localisation generally does not take place self-consistently, but requires the imposition of ad hoc rules, usually aMohr–Coulomb rheology for the brittle layer and a strain softeningrule for the viscous layer. The Mohr–Coulomb rheology formaliseslocalisation as a mathematical concept through a particular shape ofthe yield envelope and a particular choice of flow rule. The mostobvious effect of this imposition is that faults in the brittle layer havehigh angles constant with depth, so that listric and detachment faultsdo not arise naturally (Wijns et al., 2005). More subtly, ad hoc rules oflocalisationmodify the geometric and thermal evolution of the system.

3. Energy feedback effects

In order to model the spontaneous process of strain localisationwithout imposing ad hoc rules, it is necessary to shift attention fromcritical stress states to critical energy states. This is because it is theprocess of energy dissipation that gives rise to feedback effects thatlocalise strain. One way to do this is by investigating the dynamicevolution of the lithospheric strength profile using full couplingbetween continuity (e.g. conservation of mass), momentum (balanceof forces) and energy equations (Regenauer-Lieb et al., 2006). Theenergy equation describes the dynamic evolution of the energy fluxesin the system, and it is the only time-dependent equation. Conse-quently, time-dependent instabilities only arise from feedbackprocesses including the energy equation. This is a fundamentaldifference to the quasi-static Brace–Goetze approach, in which thebrittle field is entirely time independent, and likewise, the ductile fieldhas no time-dependent strength evolution. In order to overcome thisdeficiency, strain weakening laws have been proposed (e.g. Huismanset al., 2005; Lavier and Manatschal, 2006); however, these weakeninglaws are not derived self-consistently.

Our modelling approach incorporates energy feedback effects asdriving forces for instabilities. These instabilities can be caused by self-localising pressure feedback effects that are dominant in the brittleparts of the system, and by self-localising temperature feedbackeffects that are dominant in the viscous parts of the system (Fig. 2). Inthese feedback loops, any pressure or temperature variations in therock trigger feedback effects.

There are three possible sources for pressure variations ingeological systems. One is by thermal expansion associated withthermal anomalies within thematerial. Mathematically, the isentropicwork (W) of thermal expansion can be expressed as:

dWdt

¼ a DTdpdt

ð1Þ

where α is the thermal expansion coefficient and dpdt

is the resultingpressure perturbation resulting from the thermal strain, expressed asthe product of α ΔT.

Fig. 2. Energy flux feedback loop linking the energy, continuity and momentumequations. Subtle changes in pressure and temperature can lead to a transition fromelastic rheology to elasto-visco-plastic material behaviour, thereby having a large effecton the deformation of rocks. The key equations describing this transition are theconstitutive rheological equations that define the strain rate (ɛ̇) as a function of stress (σ),temperature (T), pressure (P), grain size (d) etc.

Fig. 4. Pressure variations caused by a rigid indenter pushing into softer material withnormal stress free boundary conditions (σn=0). The indenter causes the development oftwo symmetric fans of dextral and sinistral slip lines. For ideal plastic conditions withshear strength τ, the pressure variations can be calculated from the slip line theory (seeRegenauer-Lieb and Petit 1997 for quantitative solutions). From area A to area C, thepressure increases systematically by a factor of 4.1 times the yield stress (Regenauer-Lieb and Petit, 1997).

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Pressure perturbations can also result from the deformation ofrocks with minerals of different strength contrasts. This is afundamental mechanism in polycrystalline rocks under any pressureand temperature conditions. The magnitude of pressure variations isdependent on the geometry of rigid or soft inclusions, the deformationboundary conditions and rheology of the materials. For simplegeometries, pressure variations can be calculated analytically, asdone, for example, by Johnson et al. (1970) for rigid-plastic rheology,and Schmid and Podladchikov (2004) for linear viscous rheology.Pressure variations associated with an ellipse in simple shear and arigid mineral indenting softer material are shown in Figs. 3 and 4,respectively.

The third possible source of pressure variations is associated withvolume changes related to phase transitions. This is not considered inour models because it is tied to specific pressure–temperatureconditions dependent on specific rock chemistry.

Fig. 3. Pressure variations associated with material strength contrasts in a simple shearsetting. A rigid elliptic inclusion is embedded in a viscous matrix, causing low-pressure(LP) and high-pressure (HP) domains. Pressure variations are of the order of 3.5 timesthe background pressure (modified after Schmid and Podladchikov, 2004). Similarbehaviour of pressure distribution can be obtained for elastic material in contactmechanics (Johnson, 1989).

Fig. 5 shows how local pressure variations can localise failure. Here,we consider a particle loaded to a value close to critical failure. Localpressure decreases around the particle can lead to failure, whereasincreases lead to a move away from the failure envelope. Thus, failureis localised at low pressure sites, and large scale structures arise out ofa catastrophic linkage of several of these regions across the rock mass.

Self-Localising temperature feedback processes rely chiefly onsmall temperature perturbation (caused, for example, by chemicalreaction, heat of solution, radiogenic heat or phase changes) thatweakens the rock, leading to increased strain rate and shear heating.The consequences of these feedback effects can be threefold: (1) stresscan be relaxed under constant strain rate with no Localisation; (2) thedeformation mode can switch over to a pressure-sensitive instabilityas described above; and (3) the elastic energy stored in the rock masscan be transformed into ductile shear heating in zones of localisedstrain. In this case, localisation results from a short-term runaway

Fig. 5. Schematic diagram showing the effect of local pressure variations on brittlematerial in a sub-critical state. A positive (compressive) pressure perturbations leads tostabilisation, whereas negative (tensional) perturbations lead to failure.

Fig. 6. Strain rate evolution of extensional structures during early stages of a generic fully coupled crustal model (quartz rheology). Extension velocity is 1.0 cm/yr and surface heatflow is 60 mW/m2 (for rheological parameters see Table 1). Numerical results show the magnitude of strain rate after (a) 13 kyr, (b) 72 kyr, and (c) 834 kyr. The β factor for the laststage is 1.4. The elastic core formed in earlier stages where the quartz rheology was strongest and became the zone of highest energy dissipation. This led to most efficient strainlocalisation into zones of high strain rate surrounding lozenges of very low strain rate. Thus, the initially strongest layer became the weakest layer in the system.

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process through an accelerating shear heating instability, whereby thetemperature increase causes a decrease in viscosity, increased strainrate, and increased temperatures closing the loop.

The only negative feedback effects that prevent catastrophic failureare provided by (1) thermal conduction, dissipating the thermal pulsethat develops through ductile or brittle instabilities, and (2) imposed

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constant velocity boundary conditions that stabilises velocities. Thegeneral framework of the brittle and ductile feedback loops, as shownin Fig. 2, shows that the equation of state is tied to the continuityequation, and the rheology acts as the most important feedback filterwith its dependence on temperature, pressure, deviatoric stresses,grain strain, water content, activation energy and other materialparameters. The temporal evolution of the process is thus controlledinherently by the energy equation, leading to weak structures thatdeform the lithospheremore efficiently bymeans of strain localisation.

This approach is an extension of classical fluid dynamics, in whichpattern development (e.g., convection) is controlled by temperatureevolution. The important difference is that classical fluid dynamics doesnot incorporate stored energy from elasticity. The incorporation ofelastic rheology changes the fundamental energetics of the deformingbodies. The current limitation in themodel is that the energy feedbacksof the top purely brittle layer (down to ~2 km depths) are not yetconsidered. This is because at shallow depths, the surface energy ofcracks becomes increasingly important, but the current numericalscheme is unable to model this process. A more detailed description ofthe feedback loops, the equations employed and a comparison to theclassical continuum mechanics approach is described elsewhere(Regenauer-Lieb and Yuen, 2003, 2004; Kaus and Podladchikov, 2006;Regenauer-Lieb et al., 2006; Regenauer-Lieb and Yuen, in press).

4. A dynamic approach to lithospheric strength

Static shear stress quantifies the quasi-static strength of material incontinuum mechanics. The effective viscosity, defined as the shearstress divided by the associated strain rate, is the equivalent dynamicstrength measured in fluid dynamics. A shear zone would normallyhave a high shear stress andwould therefore be described as relativelystrong in a continuum mechanics sense. However, its lower viscosity(resulting from associated high strain rate) accounts for a relativelylow strength in a fluid dynamic sense. In order to compare thedynamic strength profile, which considers energy feedback mechan-isms, with the quasi-static Brace–Goetze strength profile (Fig. 1), werefer only to the classical continuummechanics shear stress definitionof lithospheric strength.

The black curve in Fig. 1 shows a snapshot of a dynamic strengthprofile for calculations that fully couple the three equations in Fig. 2.The dynamic strength is calculated shortly (130 kyr) after the onset ofextension. The dashed curve shows the equivalent static strength ofthe classical quasi-static Brace–Goetze profile for the same model.

At the early stage of deformation shown in Fig. 1, there is still somesimilarity between the dynamic strength profile and the quasi-staticBrace–Goetze profile. At later stages, the shape of the dynamic strengthcurve would be profoundly different to that of the Brace–Goetze curve(see Fig. 7 in Regenauer-Lieb and Yuen, in press). Even at this earlystage, it is obvious that the integrated strength of the lithosphere (thearea to the left of each curve) is much weaker when localisationfeedbacks are considered (Fig. 1). The top brittle crust and the areabelow 52 km already show drastic stress drops due to unloading ofelastic stresses by active, high-stress shear zones. These areas havebeen subjected to considerable weakening through the positivefeedback mechanism after the onset of visco-elasto-plastic flow.While the bottom layer is weakened through shear heating feedbackand its effect on rheology, the top layer is weakened primarily throughthermal expansion feedback. These weakening mechanisms amplifythe visco-elastic stress amplification described earlier (Kusznir, 1982).

Another significant difference between the classical Brace–Goetzeprofile and the dynamic strength profile is the fact that the latter is notapplicable in one-dimension. The effect of weakening in shear zonesimmediately leads to the focusing of stresses into the shear zones,thereby unloading the stress stored elastically in adjacent quasi-rigidblocks. This effect shows that elasticity plays a major role in a fullycoupled energy approach. Stress variations in 2D or 3D in our models

and in nature are considerably larger than those considered in a quasi-static approach (Fig. 1).

5. Self-organization: communication across brittle andductile layers

The implementation of the energy feedback approach in numericalmodels, with its concept of dynamic strength profile, has profoundimplications for understanding geodynamic processes. Small pertur-bations in a particular area can have large effects for the whole model.Conversely, large variations can decay because they become energe-tically less favourable. Therefore, the incorporation of energy feed-backs into numerical simulations leads to rich solutions for the wayfaults nucleate and propagate through the lithosphere, resulting in arich variety of extension styles.

Examples of 2D crustal extension models are presented inFigs. 6 and 7. Results show that the dynamics of extension is inti-mately linked with the development of strong elastic cores, whichcomprise regions with the highest regional stresses and where thedominant deformation mechanism is initially elastic. The strongelastic cores become zones of highest energy dissipation, which leadto strain localisation into zones of intense weakening (Fig. 6b) andultimately become high strain rate and high stress detachment zones(Regenauer-Lieb et al., 2006). These zones are strong in the continuummechanics sense and weak in the fluid dynamics sense. Multipleelastic cores can be developed depending on compositional and rhe-ological stratification.

In the early stages of deformation, the model lithosphere is loadedand ephemeral brittle fracturing propagates downwards from theweak surface (Fig. 6a). These faults progressively propagate towardsthe brittle–ductile transition (Fig. 6b), coalescing into distinct zones ofhigh strain rate (Fig. 6c). As loading continues, strain localisation alsooccurs in the ductile layer, propagating upwards towards the brittle–ductile transition (Fig. 7).

The existence of the strong elastic core is the key to understandingearly stages of energy communication across brittle and ductile layers.In the process of upward shear zone or downward brittle faultpropagation, energy in the form of heat and stored elastic energy istransferred from the termination of these localised bands to the elasticcore, thus loading this region with stored energy. Communicationbetween localisation in the upper and lower crusts takes place throughthe strong elastic band. At some point, loading of this strong layer leadsto strain localisation into well-defined shear bands characterised byhigh strain rate (Fig. 6b) and shear heating, surrounding blocks of lowstrain rate. This process transforms the elastic layer from the strongestin the system, to theweakest. In other words, strain localisation is bestdeveloped in the strongest layer, leading to high energy dissipationand effective weakening.

Ductile creep is amuchslowerprocess thanbrittle faulting; this is thereason for the delay in localisation in the weak ductile parts of thelithosphere. In the absence of significant weakening related tomicrostructural modification, such as grain size reduction or meta-morphic reactions in the fault zone, brittle faults are ephemeral. Whenductile shear zones form, the system eventually organises itself throughthe communication between fast brittle faults and slow ductile shearzones (Fig. 7). This happens when brittle faults become rooted on theductile shear zones and thus become permanent. Thus development oflong term structures is controlled by these rooted brittle faults. At thisstage, the elastic layer is preserved in well-defined elastic cores anddetachment faults are developed in zones where brittle and ductilelocalisation phenomena interact (Fig. 7b). The different rates at whichlocalised strain zonesmove implies that short timescale phenomena arecontrolled by brittle features whereas long timescale phenomena arelikely to be controlled by ductile shear zones.

An important result is that the brittle–ductile transition is notdefined by a single depth but appears in a broad transition zone in

Table 1Parameters used in numerical models

Parameter Name Value Units

Χ Shear heating efficiency 1⁎ –

κ Thermal diffusivity Q=0.7×10−6 m2s−1

F=0.7×10−6

O=0.8×10−6

αth Thermal expansion 3×10−5 K−1

cP Specific heat Q=800 J kg−1K−1

F=800O=1000

Ρ Density Q=2800 kg m−3

F=2800O=3300

V Poisson ratio 0.25 –

E Young's modulus 4.5×1010 PaA Material constant—pre-exponential

parameterQ†=1.3×10−34 Pa−n s−1

F§=7.9×10−26

O#=3.6×10−16

N Power-law exponent Q†=4 –

F§=3.1O=3.5

H Activation enthalpy Q†=135 kJ mol−1

F§=163O#=480

B Thickness of radiogenic layer 10 kmQs Surface heat flow 50/60 mWm−2

Qm Mantle heat flow 20 mWm−2

Note: Q=Quartz; F=Feldspar; O=Olivine. ⁎Chrysochoos and Belmahjoub (1992), †Hirthet al. (2001), §Shelton and Tullis (1981), #Hirth and Kohlstedt (2004).

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which brittle and ductile processes interact. Within this transitionzone, detachment zones form in the area where ductile deformationcommunicates with brittle deformation. Such detachments can

Fig. 7. Strain evolution of extensional structures in a generic fully coupled crustal model (qrheological parameters see Table 1). Numerical results show the magnitude of strain after (

initially form at relatively shallow depth (~5 km, Fig. 7b) and areprogressively exhumed during extension.

In summary, the strong elastic layer has a fundamental role incontrolling the style of extensional deformation. In cases where theolivine-dominated upper mantle is hot, the strongest elastic coredevelopswithin the crust, dominating the development of crustal corecomplexes, while the mantle responds passively (Rosenbaum et al.,2005; Regenauer-Lieb et al., 2006). In cases where the mantle is cold,the dominant elastic core shifts to a brittle–ductile transitiondeveloped in the upper mantle, resulting in mantle core complexes(Weinberg et al., 2007).

6. Discussion

6.1. The role of shear heating

Shear heating is a fundamental physical process in our models andis considered to have a key thermodynamic effect in the processes ofstrain localisation in nature, such as was found for material sciencesandmetallurgy (e.g. Coffin and Rogers,1967; Braeck and Podladchikov,2007). However, shear heating has not been unambiguously docu-mented in nature, and its role during deformation andmetamorphismremains controversial (e.g. Camacho et al., 2001; Bjørnerud andAustrheim, 2002). At small spatial and temporal scales, evidence forco-seismic shear heating and rock melting is recognised in pseudo-tachylites (Magloughlin and Spray, 1992). A larger scale (~1000 km)effect of shear heating has also been postulated, for example, for thegreat 1994 Bolivia earthquake (e.g. Kanamori et al., 1998).

Temperature rises of the order of a centigrade are capable oflocalising strain (Regenauer-Lieb and Yuen, 1998). Using examplesfrom material science, Braeck and Podladchikov (2007) propose thatshear heating at a very small scale is a fundamental instability limiting

uartz rheology). Extension velocity is 1.0 cm/y and surface heat flow is 60 mW/m2 (fora) 595 kyr (β factor=1.3) and (b) 1.28 Myr (β factor=1.7).

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the strength of all materials. Such small temperature increase isnormally not detectable in the field. In some cases, however, strainlocalisation may lead to a thermal runway during seismic events(Kelemen and Hirth, 2007) to the point of formingmelts. The difficultyin this case is that once this runway process occurs it is impossible totell whether shear heating resulted from localisation or causedlocalisation. One possible way to detect the effect of shear heating isto look at larger scale shear zones that are equivalent to the char-acteristic thermal diffusion length scale of that shear zone. Forexample, assuming a shear zone with characteristic effective strainrate (:eeff ) of 10−12 s−1, and rock diffusivity (κt) of 10−6 m2/s, thecharacteristic thermal width (l) of the shear zone will be in the orderof a kilometre (Eq. 2).

lfffiffiffiffiffiffiffijt:eeff

rð2Þ

The analyses of Braeck and Podladchikov (2007), as well as ourown models (Regenauer-Lieb et al., 2006), suggest that self-localisinginstabilities without melting must be far more common than runawaymelting instabilities. This self-localisation is very important for de-formation on geological timescales, because they ensure longevity offaults. From a geodynamic perspective it is hence important to findevidence for shear heating on a kilometre length scale.

6.2. Sensitivity to heat flux

In our previous model results (Rosenbaum et al., 2005; Regenauer-Lieb et al., 2006; Weinberg et al., 2007), we have explored the mainfeatures of extensional systems, aiming at predicting rather thanprescribing strain weakening phenomena. Here, we wish to demon-

Fig. 8. Results of numerical models showing extension of a three layer (quartz (Qz), feldsparInitial crustal thickness is 30 km. Results are calculated after 13.7 Myr extension with totconditions follow a steady state isotherm from Equations 4–31 in Turcotte and Schuberlayer=10 km; crustal conductivity=2.8 Wm−1K−1; mantle heat flow=20 mWm−2; surface twhite noise perturbations on 4% of the nodes. (a) Results for surface heat flow of 50 mWm

strate that the style of extensional features, and in turn, the extensionalmode, is strongly sensitive to small variations in heat flux.

In a classical quasi-static approach to localisation, which excludesthe energy feedback approaches, we would expect that raising theheat flow through the crust would result in a continuous trend ofthinning of the brittle layer (or layers) related to the shallowing of thebrittle–ductile transition (Ranalli andMurphy,1987). A similar effect isalso seen in our fully coupled dynamic approach, but this iscomplemented by additional, distinct, non-linear mode transitions.Fig. 8 shows the differences in results related to an increase in heatflow, equivalent to ΔTMoho of 34 °C.

Results show that the slightly hotter model (b) is stronglydominated by doming of the mantle associated with well-developedmantle detachment and pronounced Moho topography. In contrast,the slightly colder model (a) has better developed crustal detachmentand weakly developed mantle detachments, leading to an undulatingMoho, lacking a well-defined long wavelength. Interestingly, thedifference in Moho topography is opposite to the general trendobtained for higher temperature deviations, where a hot modelresulted in a flat Moho (Regenauer-Lieb et al., 2006) whereas a coldmodel was characterised by distinct mantle doming (Weinberg et al.,2007). This apparent contradiction results from the sensitivity of thesystem to interference between crustal and mantle localisationwavelengths that can be constructive or destructive.

6.3. Is the lithosphere too strong?

A key question related to continental break-up is whether or nottypical continental lithospheres are too strong to split apart unassistedby magmas (Buck, 2006). Underlying this question are roughestimates of lithospheric strengths based on Brace–Goetze strength

(Fld) and olivine (Ol)) continental lithosphere (see Table 1 for rheological parameters).al velocity of 1.2 cm/yr. The feldspar layer is highlighted in dark grey. Initial thermalt (2002). The following initial conditions are used: thickness of the heat producingemperature=280 K. The steady state isotherm is initially perturbed by 70 K amplitude−2; (b) Results for surface heat flow of 60 mWm−2.

Fig. 9. Integrated forces calculated for lithosphere subjected to extension at 1.2 cm/yrshown in Fig. 8b. The original crustal thickness of 30 km is underlain by 50 km thickolivine mantle. The surface heat flow is 60 mWm−2. Calculation is shown with andwithout feedback effects (black and dashed lines, respectively). The maximum forcecalculated for the coupled model at 2.55 ·1013 N/m is equivalent to 3.54 ·108 Pa. Thisvalue is much lower than typical forces calculated for slab pull (4.9×1013 N/m) buthigher than typical forces for ridge push (3.41 ·1012 N/m). Typical slab pull and ridgepush values are from Turcotte and Schubert (2002). Note that these values are order ofmagnitude estimates due to uncertainties in extrapolation of laboratory experiments.Nevertheless, significant weakening of the coupled versus uncoupled solutions occursindependently of absolute strength.

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profiles. Fig. 9 shows the calculated lithospheric strength measured asthe integrated force applied in order to maintain a constant verticalboundary velocity. Results are shown for a fully coupled model and atraditional model that does not take into account feedback effects. Forboth scenarios the integrated force decreases with time because of theeffect of lithospheric thinning. However, in the fully coupled model,the total strength of the lithosphere reaches a peak corresponding to aquarter of the peak reached when no feedback effects are considered.This implies that the lithosphere can rupture well below themaximum conceivable geodynamic forcing provided by slab pull.Therefore, plate tectonic processes other than subduction can have aneffect on the mobility of the continental lithosphere. For example, ithas been suggested that stresses induced by gravitational potentialenergy variations allow catastrophic weakening responsible forintraplate deformation (Coblentz and Sandiford, 1994; Sandifordet al., 2005). In a fully coupled framework, the role of weakeningthrough magmatism does not have to be invoked.

7. Conclusions

Increasingly powerful numericalmodels in 2D and 3D are revealingthe physical processes underlying extensional systems. Such modelsfall into two categories: (1) engineering-style models that use ad hocrules for strain localisation; and (2) thermodynamic-style models thatuse an energy feedback approach to derive spontaneous localisationphenomena from the supposed underlying physical processes. Theapplication of the second approach to geological problems is relativelynew (Regenauer-Lieb et al., 2001; Regenauer-Lieb and Yuen, 2003;Kaus and Podladchikov, 2006; Regenauer-Lieb et al., 2006; Braeck andPodladchikov, 2007) and requires further field-based testing. Itsapplication has produced encouraging results such as the spontaneous

development of a number of structures that match well-documentedfeatures of extensional systems, including listric and detachmentfaults, as well as exhumation of crustal and mantle core complexes. Italso indicates that the simplicity of the Brace–Goetze strength profileof the lithosphere might not appropriately define the strength of thelithosphere. When energy feedback effects are considered, the litho-sphere weakens as strain accumulates. Most intensive strain localisa-tion and energy dissipation through shear heating takes place in theinitially strongest, elastic section of the lithosphere. This processtransforms the strongest layer into the weakest. This process is whatallows for continental break-up in the absence of mantle-deriveddykes. The dynamic and structural evolution of such a system isstrongly sensitive to initial conditions such as heat flux, and the natureof the initial rheological layering of the lithosphere. Therefore, in orderto understand extensional systems it is necessary to take a multi-disciplinary approach that integrates the dynamic evolution of litho-spheric strength and strain localisation with a much deeper under-standing of the multiple roles of mafic and felsic magmatism and theability of the mantle to produce melt.

Acknowledgements

We wish to thank Mike Sandiford for comments on an earlierversion of this manuscript. K.R-L thanks the Western AustraliaGovernment for a Premier Research Fellowship.

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